v2010.10.26 - Convex Optimization

214 CHAPTER 2. CONVEX GEOMETRYConic coordinate definition (486) acquires its heritage from conditions(376) for generator membership to a smallest face. Coordinate t ⋆ v(c)=0, forexample, corresponds to unbounded µ in (376); indicating, extreme directionv cannot belong to the smallest face of cone K that contains c .2.13.12.0.2 Proof. Vector x −t ⋆ v must belong to the coneboundary ∂K by definition (486). So there must exist a nonzero vector λ thatis inward-normal to a hyperplane supporting cone K and containing x −t ⋆ v ;id est, by boundary membership relation for full-dimensional pointed closedconvex cones (2.13.2)x−t ⋆ v ∈ ∂K ⇔ ∃ λ ≠ 0 〈λ, x−t ⋆ v〉 = 0, λ ∈ K ∗ , x−t ⋆ v ∈ K (330)whereK ∗ = {w ∈ R n | 〈v , w〉 ≥ 0 for all v ∈ G(K)} (369)is the full-dimensional pointed closed convex dual cone. The set G(K) , ofpossibly infinite cardinality N , comprises generators for cone K ; e.g., itsextreme directions which constitute a minimal generating set. If x −t ⋆ vis nonzero, any such vector λ must belong to the dual cone boundary byconjugate boundary membership relationλ ∈ ∂K ∗ ⇔ ∃ x−t ⋆ v ≠ 0 〈λ, x−t ⋆ v〉 = 0, x−t ⋆ v ∈ K , λ ∈ K ∗ (331)whereK = {z ∈ R n | 〈λ, z〉 ≥ 0 for all λ ∈ G(K ∗ )} (368)This description of K means: cone K is an intersection of halfspaces whoseinward-normals are generators of the dual cone. Each and every face ofcone K (except the cone itself) belongs to a hyperplane supporting K . Eachand every vector x −t ⋆ v on the cone boundary must therefore be orthogonalto an extreme direction constituting generators G(K ∗ ) of the dual cone.To the i th extreme direction v = Γ i ∈ R n of cone K , ascribe a coordinatet ⋆ i(x)∈ R of x from definition (486). On domain K , the mapping⎡t ⋆ (x) =⎢⎣⎤t ⋆ 1(x)⎥. ⎦: R n → R N (487)t ⋆ N (x)